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# Section 8.1 Answers

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1.

1. $$\frac{1}{s^{2}}$$
2. $$\frac{1}{(s+1)^{2}}$$
3. $$\frac{b}{s^{2}-b^{2}}$$
4. $$\frac{-2s+5}{(s-1)(s-2)}$$
5. $$\frac{2}{s^{3}}$$

2.

1. $$\frac{s^{2}+2}{\left[(s-1)^{2}+1\right]\left[(s+1)^{2}+1\right]}$$
2. $$\frac{2}{s(s^{2}+4)}$$
3. $$\frac{s^{2}+8}{s(s^{2}+16)}$$
4. $$\frac{s^{2}-2}{s(s^{2}-4)}$$
5. $$\frac{4s}{(s^{2}-4)^{2}}$$
6. $$\frac{1}{s^{2}+4}$$
7. $$\frac{1}{\sqrt{2}}\frac{s+1}{s^{2}+1}$$
8. $$\frac{5s}{(s^{2}+4)(s^{2}+9)}$$
9. $$\frac{s^{3}+2s^{2}+4s+32}{(s^{2}+4)(s^{2}+16)}$$

4.

1. $$f(3-)=-1,\: f(3)=f(3+)=1$$
2. $$f(1-)=3,\: f(1)=4,\: f(1+)=1$$
3. $$f\left(\frac{\pi }{2}-\right) =1,\: f\left(\frac{\pi }{2} \right) = f\left(\frac{\pi }{2}+ \right)=2,\: f(\pi -)=0,\: f(\pi )=f(\pi +)=-1$$
4. $$f(1−) = 1,\: f(1) = 2,\: f(1+) = 1,\: f(2−) = 0,\: f(2) = 3,\: f(2+) = 6$$

5.

1. $$\frac{1-e^{-(s+1)}}{s+1}+\frac{e^{-(s+2)}}{s+2}$$
2. $$\frac{1}{s}+e^{-4s}\left(\frac{1}{s^{2}}+\frac{3}{s}\right)$$
3. $$\frac{1-e^{-s}}{s^{2}}$$
4. $$\frac{1-e^{-(s-1)}}{(s-1)^{2}}$$

7. $$\mathcal{L}(e^{\lambda t}\cos\omega t)=\frac{(s-\lambda )^{2}-\omega^{2}}{((s-\lambda )^{2}+\omega^{2})^{2}}\quad\mathcal{L}(e^{\lambda t}\sin\omega t)=\frac{2\omega (s-\lambda )}{((s-\lambda )^{2}+\omega ^{2})^{2}}$$

15.

1. $$\tan ^{-1}\frac{\omega }{s},\quad s>0$$
2. $$\frac{1}{2}\ln\frac{s^{2}}{s^{2}+\omega ^{2}},\quad s>0$$
3. $$\ln\frac{s-b}{s-a},\quad s>\text{max}(a,b)$$
4. $$\frac{1}{2}\ln\frac{s^{2}}{s^{2}-1},\quad s>1$$
5. $$\frac{1}{4}\ln\frac{s^{2}}{s^{2}-4},\quad s>2$$

18.

1. $$\frac{1}{s^{2}}\tanh\frac{s}{2}$$
2. $$\frac{1}{s}\tanh\frac{s}{4}$$
3. $$\frac{1}{s^{2}+1}\coth\frac{\pi s}{2}$$
4. $$\frac{1}{(s^{2}+1)(1-e^{-\pi s})}$$